The.Algorithm.Design.Manual.Springer-Verlag.1998

The Partition Problem
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The Partition Problem
Suppose that three workers are given the task of scanning through a shelf of books in search of a given piece of information. To
get the job done fairly and efficiently, the books are to be partitioned among the three workers. To avoid the need to rearrange
the books or separate them into piles, it would be simplest to divide the shelf into three regions and assign each region to one
worker.
But what is the fairest way to divide the shelf up? If each book is the same length, say 100 pages, the job is pretty easy. Just
partition the books into equal-sized regions,
so that everyone has 300 pages to deal with.
But what if the books are not the same length? Suppose we used the same partition when the book sizes looked like this:
I, for one, would volunteer to take the first section, with only 600 pages to scan, instead of the last one, with 2,400 pages. The
fairest possible partition for this shelf would be
where the largest job is only 1,700 pages and the smallest job 1,300.
In general, we have the following problem:
Input: A given arrangement S of non-negative numbers and an integer k.
Output: Partition S into k ranges, so as to minimize the maximum sum over all the ranges. This so-called linear partition
problem arises often in parallel processing, since we seek to balance the work done across processors so as to minimize the
total elapsed run time. Indeed, the war story of Section revolves around a botched solution to this problem.
Stop for a few minutes and try to find an algorithm to solve the linear partition problem.
The beginning algorist might suggest a heuristic as the most natural approach to solve the partition problem. Perhaps they
would compute the average size of a partition, , and then try to insert the dividers so as to come close to this average.
However, such heuristic methods are doomed to fail on certain inputs, because they do not systematically evaluate all
possibilities.
Instead, consider a recursive, exhaustive search approach to solving this problem. Notice that the kth partition starts right after
we placed the (k-1)st divider. Where can we place this last divider? Between the ith and (i+1)st elements for some i, where
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